Background technology will be described with reference to FIGS. 1 and 2. The details of the background technology are disclosed in Non-Patent Document 1.
FIG. 1 depicts an example of a configuration of a communication apparatus using BICM-ID (Bit-Interleaved Coded Modulation with Iterative Decoding) using an extended mapping, which shows configurations of a transmitting unit and a receiving unit.
The communication apparatus may be, e.g., a transmitter having only a function of the transmitting unit or a receiver having only a function of the receiving unit or a transceiver having both functions of the transmitting unit and the receiving unit.
The transmitting unit includes an encoder 1, an interleaver 2, a mapper 3, a D/A (digital to analog) converter 4, a transmission RF (radio frequency) section 5, and a transmission antenna 6.
The receiving unit includes a reception antenna 11, a reception RF section 12, an A/D (analog to digital) converter 13, a demapper 14, a deinterleaver 15, a decoder 16, and an interleaver 17.
FIG. 2 shows an example of the extended mapping, wherein the extended mapping is performed on a 5-bit sequence B (b0, b1, b2, b3, b4) in QPSK (Quadrature Phase Shift Keying).
In the present example, among four symbols in an I-Q plane, a symbol S0 is assigned with bit sequences B0˜B7 (10011, 10110, 01011, 10101, 01110, 01101, 11111, 00111); a symbol S1 is assigned with bit sequences B8˜B15 (00100, 00001, 00010, 11010, 11001, 10000, 01000, 11100); a symbol S2 is assigned with bit sequences B16˜B23 (10111, 01111, 00110, 11101, 00101, 00011, 11110, 11011); and a symbol S3 is assigned with bit sequences B24˜B31 (10010, 00000, 11000, 10001, 01010, 10100, 01001, 01100).
Hereinafter, the details regarding a problem in the background technology will be mainly described.
In the transmitting unit, a transmission bit is inputted to the encoder 1 and the encoder 1 carries out encoding (e.g., turbo coding) of the transmission bit. The encoded signal is inputted to the mapper 3 via the interleaver 2, and the mapper 3 carries out a mapping process. Here, as a method of mapping, an extended mapping is used to assign a plurality of bit sequences to a single symbol point, of which example is shown in FIG. 2. The signal mapped by the mapper 3 is converted from a digital signal to an analog signal by the D/A converter 4. The converted signal is transmitted by radio to the transmission antenna 6 via the transmission RF section 5 for the transmission process.
In the receiving unit, a signal received from the reception antenna 11 (the signal transmitted from the transmitting unit) is converted to a base band signal by the reception RF section 12 and the base band signal is inputted to the A/D converter 13. This signal is converted from an analog signal to a digital signal by the A/D converter 13 and the converted signal is inputted to the demapper 14. The demapper 14 calculates a log likelihood ratio (LLR) of every bit, based on a received signal (the signal from the A/D converter 13), a noise power included in the received signal, and advance information obtained through the interleaver 17 from decoder 16. This calculation will be later described in detail.
The LLR, calculated by the demapper 14, is inputted to the decoder 16 via the deinterleaver 15. The decoder 16 carries out decoding based on the inputted LLR, thereby calculating a second LLR. The second LLR calculated by the decoder 16 is inputted (fed back) to the demapper 14 via the interleaver 17 and the demapper 14, again, calculates the LLR based on the received signal, the noise power, and the advance information which is fed back.
By iteratively performing the above process, a good decoding result can be satisfactorily acquired. The advance information which is fed back is assumed as zero in a first iteration (in a first operation of the demapper 14).
The LLR calculation by the demapper 14 will be described now.
When a N-bit (N is an integer equal to or more than 1 or 2) sequence B (b0, b1, . . . , bN-1) is assigned to M (M is an integer equal to or more than 1 or 2) number of symbol points Sk, the LLR outputted from the demapper 14 is considered.
Eq. (1) is established when the received signal is y, i-th (i=0, 1, . . . , N−1) bit is bi, and the LLR for bi is L(bi):
                                                                        L                ⁡                                  (                                      b                    i                                    )                                            =                            ⁢                              log                ⁢                                                      p                    ⁢                                          (                                                                        b                          i                                                =                                                  0                          ⁢                                                      ❘                                                    ⁢                          y                                                                    )                                                                            p                    ⁡                                          (                                                                        b                          i                                                =                                                  1                          ⁢                                                      ❘                                                    ⁢                          y                                                                    )                                                                                                                                              =                            ⁢                              log                ⁢                                                                            p                      ⁡                                              (                                                                              y                            ⁢                                                          ❘                                                        ⁢                                                          b                              i                                                                                =                          0                                                )                                                              ⁢                                          p                      ⁡                                              (                                                                              b                            i                                                    =                          0                                                )                                                              ⁢                                          /                                        ⁢                                          p                      ⁡                                              (                        y                        )                                                                                                                        p                      ⁡                                              (                                                                              y                            ⁢                                                          ❘                                                        ⁢                                                          b                              i                                                                                =                          1                                                )                                                              ⁢                                          p                      ⁡                                              (                                                                              b                            i                                                    =                          1                                                )                                                              ⁢                                          /                                        ⁢                                          p                      ⁡                                              (                        y                        )                                                                                                                                                                    =                            ⁢                                                log                  ⁢                                                            p                      ⁡                                              (                                                                              y                            ⁢                                                          ❘                                                        ⁢                                                          b                              i                                                                                =                          0                                                )                                                                                    p                      ⁡                                              (                                                                              y                            ⁢                                                          ❘                                                        ⁢                                                          b                              i                                                                                =                          1                                                )                                                                                            +                                  log                  ⁢                                                                                    p                        ⁢                                                  (                                                                                    b                              i                                                        =                            0                                                    )                                                                                            p                        ⁡                                                  (                                                                                    b                              i                                                        =                            1                                                    )                                                                                      .                                                                                                          Eq        .                                  ⁢                  (          1          )                    
As will be later described, a first term in the last right-hand side of Eq. (1) is the LLR obtained from the other bits except the i-th bit, which is referred to as external information, Le(bi). A second term in the last right-hand side of Eq. (1) is the LLR obtained from a prior probability of the i-th bit, which is referred to as La(bi).
By doing so, Eq. (1) becomes Eq. (2) and is transformed to Eq. (3):L(bi)=Le(bi)+La(bi)  Eq. (2)andLe(bi)=L(bi)−La(bi).  Eq. (3)
The demapper 14 outputs the LLR which is a result of Eq. (3).
Herein, a numerator p(y|bi=0) in the first term in the last right-hand side of Eq. (1) is considered.
The p(y|bi=0) is a probability of the received signal becoming y when given bi=0. The p(y|bi=0) is expressed as p(y|Bn)p(Bn|bi=0), wherein the p(y|Bn) is a probability of becoming y when given Bn, and the p(Bn|bi=0) is a probability of becoming a bit sequence Bn when given bi=0. As all bit sequences Bn are considered, Eq. (4) is established:
                                          p            ⁡                          (                                                y                  |                                      b                    i                                                  =                0                            )                                =                                    ∑                                                                    B                    n                                    |                                                            b                      i                                        ⁡                                          (                                              B                        n                                            )                                                                      =                0                                      ⁢                                                  ⁢                                          p                ⁡                                  (                                      y                    |                                          B                      n                                                        )                                            ⁢                              p                ⁡                                  (                                                                                    B                        n                                            |                                              b                        i                                                              =                    0                                    )                                                                    ,                            Eq        .                                  ⁢                  (          4          )                    wherein bi(Bn) represents ith bit of the bit sequence Bn.
Similarly, Eq. (5) is established with respect to a denominator p(y|bi=1) in the first term in the last right-hand side of Eq. (1):
                              p          ⁡                      (                                          y                |                                  b                  i                                            =              1                        )                          =                              ∑                                                            B                  n                                |                                                      b                    i                                    ⁡                                      (                                          B                      n                                        )                                                              =              1                                ⁢                                          ⁢                                    p              ⁡                              (                                  y                  |                                      B                    n                                                  )                                      ⁢                                          p                ⁡                                  (                                                                                    B                        n                                            |                                              b                        i                                                              =                    1                                    )                                            .                                                          Eq        .                                  ⁢                  (          5          )                    
Accordingly, the first term in the last right-hand side of Eq. (1) becomes Eq. (6):
                                                                                          L                  e                                ⁡                                  (                                      b                    i                                    )                                            =                            ⁢                              log                ⁢                                                      p                    ⁡                                          (                                                                        y                          |                                                      b                            i                                                                          =                        0                                            )                                                                            p                    ⁡                                          (                                                                        y                          |                                                      b                            i                                                                          =                        1                                            )                                                                                                                                              =                            ⁢                              log                ⁢                                                                                                    ∑                                                                                                            B                              n                                                        |                                                                                          b                                i                                                            ⁡                                                              (                                                                  B                                  n                                                                )                                                                                                              =                          0                                                                    ⁢                                                                                          ⁢                                              p                        ⁢                                                  (                                                      y                            |                                                          B                              n                                                                                )                                                ⁢                                                  p                          ⁡                                                      (                                                                                                                            B                                  n                                                                |                                                                  b                                  i                                                                                            =                              0                                                        )                                                                                                                                                              ∑                                                                                                            B                              n                                                        |                                                                                          b                                i                                                            ⁡                                                              (                                                                  B                                  n                                                                )                                                                                                              =                          1                                                                    ⁢                                                                                          ⁢                                                                        p                          ⁡                                                      (                                                          y                              |                                                              B                                n                                                                                      )                                                                          ⁢                                                  p                          ⁡                                                      (                                                                                                                            B                                  n                                                                |                                                                  b                                  i                                                                                            =                              1                                                        )                                                                                                                                .                                                                                        Eq        .                                  ⁢                  (          6          )                    
Regarding the p(y|Bn) in Eq. (6), considering that Gaussian noises of variances σI2 and σQ2 are added to I and Q, respectively in the process of transferring the bit sequence Bn and going on to the received signal y, Eq. (7) is derived:
                                          p            ⁡                          (                              y                |                                  B                  n                                            )                                =                                    1                                                2                  ⁢                                      πσ                    I                    2                                                                        ⁢                          exp              ⁡                              (                                  -                                                                                    (                                                  Re                          ⁡                                                      (                                                          y                              -                                                              B                                n                                                                                      )                                                                          )                                            2                                                              2                      ⁢                                              σ                        I                        2                                                                                            )                                      ⁢                          1                                                2                  ⁢                                      πσ                    Q                    2                                                                        ⁢                          exp              ⁡                              (                                  -                                                                                    (                                                  Im                          ⁡                                                      (                                                          y                              -                                                              B                                n                                                                                      )                                                                          )                                            2                                                              2                      ⁢                                              σ                        Q                        2                                                                                            )                                                    ,                            Eq        .                                  ⁢                  (          7          )                    wherein the notation “y−Bn” means positional difference between y and Bn (to be precise, Sk assigned with Bn) in mapping space, as shown in FIG. 2.
Further, Eq. (8) is derived when σI2=σQ2 and σI2+σQ2=2σI2=2σQ2=σ2:
                              p          ⁡                      (                          y              |                              B                n                                      )                          =                              1                          πσ              2                                ⁢                                    exp              ⁡                              (                                  -                                                                                                                                      y                          -                                                      B                            n                                                                                                                      2                                                              σ                      2                                                                      )                                      .                                              Eq        .                                  ⁢                  (          8          )                    
The p(Bn|bi=0) in Eq. (6) is a probability of becoming a bit sequence Bn when given bi=0 and it is expressed as a product of prior probabilities of the other bits except bi among the bits composing the bit sequence Bn. When j-th (j=0, 1, . . . , N−1) bit of the bit sequence Bn is bj(Bn), Eq. (9) is established:
                                                        b              j                        ⁡                          (                              B                n                            )                                ∈                      {                          0              ,              1                        }                          ,                                  ⁢                              p            ⁡                          (                                                                    B                    n                                    |                                      b                    i                                                  =                0                            )                                =                                    ∏                              j                ≠                i                                      ⁢                                                  ⁢                                          p                ⁡                                  (                                                            b                      j                                        =                                                                  b                        j                                            ⁡                                              (                                                  B                          n                                                )                                                                              )                                            .                                                          Eq        .                                  ⁢                  (          9          )                    
Here, p(bj=bj(Bn)) is considered.
When La(bj) is applied as prior information, the second term in the last right-hand side of Eq. (1) is changed to Eq. (10) and further to Eq. (11):
                                                        L              a                        ⁡                          (                              b                j                            )                                =                      log            ⁢                                          p                ⁡                                  (                                                            b                      j                                        =                    0                                    )                                                            p                ⁡                                  (                                                            b                      j                                        =                    1                                    )                                                                    ⁢                                  ⁢        and                            Eq        .                                  ⁢                  (          10          )                                                              p            ⁡                          (                                                b                  j                                =                0                            )                                            p            ⁡                          (                                                b                  j                                =                1                            )                                      =                              exp            ⁡                          (                                                L                  a                                ⁡                                  (                                      b                    j                                    )                                            )                                .                                    Eq        .                                  ⁢                  (          11          )                    
Furthermore, Eqs. (12) and (13) are established based on the relation of p(bj=0)+p(bj=1)=1:
                                          p            ⁡                          (                                                b                  j                                =                0                            )                                =                                    exp              ⁡                              (                                                      L                    a                                    ⁡                                      (                                          b                      j                                        )                                                  )                                                    1              +                              exp                ⁡                                  (                                                            L                      a                                        ⁡                                          (                                              b                        j                                            )                                                        )                                                                    ⁢                                  ⁢        and                            Eq        .                                  ⁢                  (          12          )                                                  p          ⁡                      (                                          b                j                            =              1                        )                          =                              1                          1              +                              exp                ⁡                                  (                                                            L                      a                                        ⁡                                          (                                              b                        j                                            )                                                        )                                                              .                                    Eq        .                                  ⁢                  (          13          )                    
Eq. (14) is derived by using these Eqs. (12) and (13), and Eq. (9) becomes Eq. (15):
                                          p            ⁡                          (                                                b                  j                                =                                                      b                    j                                    ⁡                                      (                                          B                      n                                        )                                                              )                                =                                    exp              ⁡                              (                                                      -                                                                  b                        j                                            ⁡                                              (                                                  B                          n                                                )                                                                              ⁢                                                            L                      a                                        ⁡                                          (                                              b                        j                                            )                                                                      )                                                    1              +                              exp                ⁡                                  (                                      -                                                                  L                        a                                            ⁡                                              (                                                  b                          j                                                )                                                                              )                                                                    ⁢                                  ⁢        and                            Eq        .                                  ⁢                  (          14          )                                                                                            p                ⁡                                  (                                                                                    B                        n                                            |                                              b                        i                                                              =                    0                                    )                                            =                            ⁢                                                ∏                                      j                    ≠                    i                                                  ⁢                                                                  ⁢                                  p                  ⁡                                      (                                                                  b                        j                                            =                                                                        b                          j                                                ⁡                                                  (                                                      B                            n                                                    )                                                                                      )                                                                                                                          =                            ⁢                                                ∏                                      j                    ≠                    i                                                  ⁢                                                                  ⁢                                                                            exp                      ⁡                                              (                                                                              -                                                                                          b                                j                                                            ⁡                                                              (                                                                  B                                  n                                                                )                                                                                                              ⁢                                                                                    L                              a                                                        ⁡                                                          (                                                              b                                j                                                            )                                                                                                      )                                                                                    1                      +                                              exp                        ⁡                                                  (                                                      -                                                                                          L                                a                                                            ⁡                                                              (                                                                  b                                  j                                                                )                                                                                                              )                                                                                                      .                                                                                        Eq        .                                  ⁢                  (          15          )                    
Similar to Eq. (15), another equation for p(Bn|bi=1) is also established.
Finally Eq. (16a) is obtained from Eqs. (6), (8), (15) and the other equation for p(Bn|bi=1), where the summation (Σ) in the numerator is adopted among bj(Bb) of 0, and the summation in the denominator is adopted among bj(Bn) of 1:
                                                                                          L                  e                                ⁡                                  (                                      b                    i                                    )                                            =                            ⁢                              log                ⁢                                                      p                    ⁡                                          (                                                                        y                          |                                                      b                            i                                                                          =                        0                                            )                                                                            p                    ⁡                                          (                                                                        y                          |                                                      b                            i                                                                          =                        1                                            )                                                                                                                                              =                            ⁢                              log                ⁢                                                                            ∑                                                                                                    B                            n                                                    |                                                                                    b                              i                                                        ⁡                                                          (                                                              B                                n                                                            )                                                                                                      =                        0                                                              ⁢                                                                                  ⁢                                                                  1                                                  πσ                          2                                                                    ⁢                                              exp                        ⁡                                                  (                                                      -                                                                                                                                                                                                  y                                    -                                                                          B                                      n                                                                                                                                                                        2                                                                                            σ                                2                                                                                                              )                                                                    ⁢                                                                        ∏                                                      j                            ≠                            i                                                                          ⁢                                                                                                  ⁢                                                                              exp                            ⁡                                                          (                                                                                                -                                                                                                            b                                      j                                                                        ⁡                                                                          (                                                                              B                                        n                                                                            )                                                                                                                                      ⁢                                                                                                      L                                    a                                                                    ⁡                                                                      (                                                                          b                                      j                                                                        )                                                                                                                              )                                                                                                            1                            +                                                          exp                              ⁡                                                              (                                                                  -                                                                                                            L                                      a                                                                        ⁡                                                                          (                                                                              b                                        j                                                                            )                                                                                                                                      )                                                                                                                                                                                                                                    ∑                                                                                                    B                            n                                                    |                                                                                    b                              i                                                        ⁡                                                          (                                                              B                                n                                                            )                                                                                                      =                        1                                                              ⁢                                                                                  ⁢                                                                  1                                                  πσ                          2                                                                    ⁢                                              exp                        ⁡                                                  (                                                      -                                                                                                                                                                                                  y                                    -                                                                          B                                      n                                                                                                                                                                        2                                                                                            σ                                2                                                                                                              )                                                                    ⁢                                                                        ∏                                                      j                            ≠                            i                                                                          ⁢                                                                                                  ⁢                                                                              exp                            ⁡                                                          (                                                                                                -                                                                                                            b                                      j                                                                        ⁡                                                                          (                                                                              B                                        n                                                                            )                                                                                                                                      ⁢                                                                                                      L                                    a                                                                    ⁡                                                                      (                                                                          b                                      j                                                                        )                                                                                                                              )                                                                                                            1                            +                                                          exp                              ⁡                                                              (                                                                  -                                                                                                            L                                      a                                                                        ⁡                                                                          (                                                                              b                                        j                                                                            )                                                                                                                                      )                                                                                                                                                                                                                                                                                =                            ⁢                              log                ⁢                                                                            ∑                                                                                                    B                            n                                                    |                                                                                    b                              i                                                        ⁡                                                          (                                                              B                                n                                                            )                                                                                                      =                        0                                                              ⁢                                                                                  ⁢                                                                  exp                        ⁡                                                  (                                                      -                                                                                                                                                                                                  y                                    -                                                                          B                                      n                                                                                                                                                                        2                                                                                            σ                                2                                                                                                              )                                                                    ⁢                                                                        ∏                                                      j                            ≠                            i                                                                          ⁢                                                                                                  ⁢                                                  exp                          ⁡                                                      (                                                                                          -                                                                                                      b                                    j                                                                    ⁡                                                                      (                                                                          B                                      n                                                                        )                                                                                                                              ⁢                                                                                                L                                  a                                                                ⁡                                                                  (                                                                      b                                    j                                                                    )                                                                                                                      )                                                                                                                                                                          ∑                                                                                                    B                            n                                                    |                                                                                    b                              i                                                        ⁡                                                          (                                                              B                                n                                                            )                                                                                                      =                        1                                                              ⁢                                                                                  ⁢                                                                  exp                        ⁡                                                  (                                                      -                                                                                                                                                                                                  y                                    -                                                                          B                                      n                                                                                                                                                                        2                                                                                            σ                                2                                                                                                              )                                                                    ⁢                                                                        ∏                                                      j                            ≠                            i                                                                          ⁢                                                                                                  ⁢                                                  exp                          ⁡                                                      (                                                                                          -                                                                                                      b                                    j                                                                    ⁡                                                                      (                                                                          B                                      n                                                                        )                                                                                                                              ⁢                                                                                                L                                  a                                                                ⁡                                                                  (                                                                      b                                    j                                                                    )                                                                                                                      )                                                                                                                                                                                                                      =                            ⁢                              log                ⁢                                                                            ∑                                                                                                    B                            n                                                    |                                                                                    b                              i                                                        ⁡                                                          (                                                              B                                n                                                            )                                                                                                      =                        0                                                              ⁢                                                                                  ⁢                                                                  exp                        ⁡                                                  (                                                      -                                                                                                                                                                                                  y                                    -                                                                          B                                      n                                                                                                                                                                        2                                                                                            σ                                2                                                                                                              )                                                                    ⁢                                                                        ∏                          j                                                ⁢                                                                                                  ⁢                                                                              exp                            ⁡                                                          (                                                                                                -                                                                                                            b                                      j                                                                        ⁡                                                                          (                                                                              B                                        n                                                                            )                                                                                                                                      ⁢                                                                                                      L                                    a                                                                    ⁡                                                                      (                                                                          b                                      j                                                                        )                                                                                                                              )                                                                                /                                                      exp                            ⁡                                                          (                                                                                                -                                                                                                            b                                      i                                                                        ⁡                                                                          (                                                                              B                                        n                                                                            )                                                                                                                                      ⁢                                                                                                      L                                    a                                                                    ⁡                                                                      (                                                                          b                                      i                                                                        )                                                                                                                              )                                                                                                                                                                                                      ∑                                                                                                    B                            n                                                    |                                                                                    b                              i                                                        ⁡                                                          (                                                              B                                n                                                            )                                                                                                      =                        1                                                              ⁢                                                                                  ⁢                                                                  exp                        ⁡                                                  (                                                      -                                                                                                                                                                                                  y                                    -                                                                          B                                      n                                                                                                                                                                        2                                                                                            σ                                2                                                                                                              )                                                                    ⁢                                                                        ∏                          j                                                ⁢                                                                                                  ⁢                                                                              exp                            ⁡                                                          (                                                                                                -                                                                                                            b                                      j                                                                        ⁡                                                                          (                                                                              B                                        n                                                                            )                                                                                                                                      ⁢                                                                                                      L                                    a                                                                    ⁡                                                                      (                                                                          b                                      j                                                                        )                                                                                                                              )                                                                                /                                                      exp                            ⁡                                                          (                                                                                                -                                                                                                            b                                      i                                                                        ⁡                                                                          (                                                                              B                                        n                                                                            )                                                                                                                                      ⁢                                                                                                      L                                    a                                                                    ⁡                                                                      (                                                                          b                                      i                                                                        )                                                                                                                              )                                                                                                                                                                                                                                                  =                            ⁢                              log                ⁢                                                                                                    ∑                                                                                                            B                              n                                                        |                                                                                          b                                i                                                            ⁡                                                              (                                                                  B                                  n                                                                )                                                                                                              =                          0                                                                    ⁢                                                                                          ⁢                                                                        exp                          ⁡                                                      (                                                          -                                                                                                                                                                                                              y                                      -                                                                              B                                        n                                                                                                                                                                                  2                                                                                                  σ                                  2                                                                                                                      )                                                                          ⁢                                                                              ∏                            j                                                    ⁢                                                                                                          ⁢                                                      exp                            ⁡                                                          (                                                                                                -                                                                                                            b                                      j                                                                        ⁡                                                                          (                                                                              B                                        n                                                                            )                                                                                                                                      ⁢                                                                                                      L                                    a                                                                    ⁡                                                                      (                                                                          b                                      j                                                                        )                                                                                                                              )                                                                                                                                                                                          ∑                                                                                                            B                              n                                                        |                                                                                          b                                i                                                            ⁡                                                              (                                                                  B                                  n                                                                )                                                                                                              =                          1                                                                    ⁢                                                                                          ⁢                                                                        exp                          ⁡                                                      (                                                          -                                                                                                                                                                                                              y                                      -                                                                              B                                        n                                                                                                                                                                                  2                                                                                                  σ                                  2                                                                                                                      )                                                                          ⁢                                                                              ∏                            j                                                    ⁢                                                                                                          ⁢                                                      exp                            ⁡                                                          (                                                                                                -                                                                                                            b                                      j                                                                        ⁡                                                                          (                                                                              B                                        n                                                                            )                                                                                                                                      ⁢                                                                                                      L                                    a                                                                    ⁡                                                                      (                                                                          b                                      j                                                                        )                                                                                                                              )                                                                                                                                                            .                                                                                        Eq        .                                  ⁢                  (                      16            ⁢                                                  ⁢            a                    )                    
When the number of iteration is 1 (first iteration), Eq. (16a) should be replaced by Eq. (16b):
                                                                                          L                  e                                ⁡                                  (                                      b                    i                                    )                                            =                            ⁢                              log                ⁢                                                      p                    ⁡                                          (                                                                        y                          |                                                      b                            i                                                                          =                        0                                            )                                                                            p                    ⁡                                          (                                                                        y                          |                                                      b                            i                                                                          =                        1                                            )                                                                                                                                              =                            ⁢                              log                ⁢                                                                                                    ∑                                                                                                            B                              n                                                        |                                                                                          b                                i                                                            ⁡                                                              (                                                                  B                                  n                                                                )                                                                                                              =                          0                                                                    ⁢                                                                                          ⁢                                              exp                        ⁡                                                  (                                                      -                                                                                                                                                                                                  y                                    -                                                                          B                                      n                                                                                                                                                                        2                                                                                            σ                                2                                                                                                              )                                                                                                                                    ∑                                                                                                            B                              n                                                        |                                                                                          b                                i                                                            ⁡                                                              (                                                                  B                                  n                                                                )                                                                                                              =                          1                                                                    ⁢                                                                                          ⁢                                              exp                        ⁡                                                  (                                                      -                                                                                                                                                                                                  y                                    -                                                                          B                                      n                                                                                                                                                                        2                                                                                            σ                                2                                                                                                              )                                                                                                      .                                                                                        Eq        .                                  ⁢                  (                      16            ⁢                                                  ⁢            b                    )                    
From the above, the demapper 14 performs an exponential operation and calculates a sum total for not only bit sequences but also every bit assigned to each of the bit sequences. Further, the demapper 14 derives the denominator and the numerator from the results and performs logarithmic calculations of them.
However, in executing the repetition process in BICM-ID, in order to calculate the LLR, the demapper 14 needs to perform the exponential operation and the log operation for each bit of each bit sequence. Accordingly, the operation amount becomes enormously large.
To reduce the operation amount, a MAX-LOG approximation such as Eq. (17) is used:
                              log          ⁡                      (                                          ∑                                  j                  =                  1                                k                            ⁢                                                          ⁢                              exp                ⁡                                  (                                      a                    j                                    )                                                      )                          ≡                              max            ⁡                          (                                                a                  1                                ,                                  a                  2                                ,                                  a                  3                                ,                …                ⁢                                                                  ,                                  a                  k                                            )                                .                                    Eq        .                                  ⁢                  (          17          )                    
Using Eq. (17), Eq. (16b) is simplified into:
                                                                                          L                  e                                ⁡                                  (                                      b                    i                                    )                                            =                            ⁢                              log                ⁢                                                                            ∑                                                                                                    B                            n                                                    |                                                                                    b                              i                                                        ⁡                                                          (                                                              B                                n                                                            )                                                                                                      =                        0                                                              ⁢                                                                                  ⁢                                          exp                      ⁡                                              (                                                  -                                                                                                                                                                                      y                                  -                                                                      B                                    n                                                                                                                                                              2                                                                                      σ                              2                                                                                                      )                                                                                                                        ∑                                                                                                    B                            n                                                    |                                                                                    b                              i                                                        ⁡                                                          (                                                              B                                n                                                            )                                                                                                      =                        1                                                              ⁢                                                                                  ⁢                                          exp                      ⁡                                              (                                                  -                                                                                                                                                                                      y                                  -                                                                      B                                    n                                                                                                                                                              2                                                                                      σ                              2                                                                                                      )                                                                                                                                                                    ≈                            ⁢                                                                    max                                                                                            B                          n                                                |                                                                              b                            i                                                    ⁡                                                      (                                                          B                              n                                                        )                                                                                              =                      0                                                        ⁢                                      (                                          -                                                                                                                                                              y                              -                                                              B                                n                                                                                                                                          2                                                                          σ                          2                                                                                      )                                                  -                                                      max                                                                                            B                          n                                                |                                                                              b                            i                                                    ⁡                                                      (                                                          B                              n                                                        )                                                                                              =                      1                                                        ⁢                                                            (                                              -                                                                                                                                                                          y                                -                                                                  B                                  n                                                                                                                                                    2                                                                                σ                            2                                                                                              )                                        .                                                                                                          Eq        .                                  ⁢                  (          18          )                    
However, in the extended mapping shown in FIG. 2, bit sequences giving maximum value in the respective first and second term in the last right-hand side of Eq. (18) are equal because such bit sequences belong to the same symbol, so that the LLR becomes zero. Therefore, the LLR outputted from the demapper 14 becomes zero for any bit. For this reason, the decoder 16 cannot obtain the prior information and the LLR is not changed and stays zero even when the processing is repeated many times.